Manuscript XIV-F@1 · XIV-F.tex · proposition · bae5a40b

Proposition: Six regular constraints and a quadratic final constraint

F2: ambient space and the intrinsic conormal matching › First new calculation: the regular stage and its last equation · lines 920–964 · proof included or linked

XIV-F@1#prop:regular, XIV-F@1#eq:Sparameter, XIV-F@1#eq:qR-on-S

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Near the retained point, $S_{6}$ is a smooth codimension-six submanifold of $W$ . It has the local parameterization

z y b = a, = a + α (1, n_{1}), = y + δ (1, n_{2}), α > 0, δ > 0, n_{1}, n_{2} \in S^{2} .

On this manifold,

q_{R} = α δ ∣ n_{1} - n_{2} ∣^{2} .

Consequently $N$ is the angular diagonal $n_{1} = n_{2}$ in $S_{6}$ , with $β = α + δ$ .

Proof. The differentials $d q_{1}, d q_{2}$ are independent at the retained points. The four components of $d (a - z)$ are independent of them because the cone equations have no $z$ component. Thus the six equations define a smooth codimension-six manifold.

The chronology and null equations uniquely write the two successive nonzero future null segments as $α (1, n_{1})$ and $δ (1, n_{2})$ . Their time components and normalized spatial directions give a smooth inverse to (eq:Sparameter), so this is a local parameterization.

The sum of the two null segments has Minkowski square

2 α δ (1 - n_{1} \cdot n_{2}) = α δ ∣ n_{1} - n_{2} ∣^{2} .

It is zero exactly when their unit spatial directions agree.

Displayed formulas in this passage

XIV-F@1#eq:Sparameter · line 924
$z y b = a, = a + α (1, n_{1}), = y + δ (1, n_{2}), α > 0, δ > 0, n_{1}, n_{2} \in S^{2} .$
XIV-F@1#eq:qR-on-S · line 936
$q_{R} = α δ ∣ n_{1} - n_{2} ∣^{2} .$
unlabeled · line 959
$2 α δ (1 - n_{1} \cdot n_{2}) = α δ ∣ n_{1} - n_{2} ∣^{2} .$

Proposition: Six regular constraints and a quadratic final constraint

Displayed formulas in this passage

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